lecture 6. 2.3 coaxial line advantage of coaxial design: little electromagnetic leakage outside the...
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Lecture 6
2.3 Coaxial line
Advantage of coaxial design: little electromagnetic leakage outside the shield and a good choice for carrying weak signals not tolerating interference from the environment or for higher electrical signals not being allowed to radiate or couple into adjacent structures or circuits.
Common application: video and CATV distribution, RF and microwave transmission, and computer and instrumentation data connections。
The transverse fields satisfy Laplace equations, i.e.
Boundary conditions
TEM mode in coaxial cables
and
Equivalent to the static case, electric field , here the electric potential also satisfies Laplace equation
(in Cartesian coordinate )
or (in cylindrical coordinate)
2.3 Rectangular waveguide
Closed waveguide, propagate Transverse electric (TE) and/or transverse magnetic (TM) modes.
expansion
HiE
EiH
)(
)(
)(
)(
22
22
22
22
yH
zxE
kki
y
xH
zyE
kki
x
xH
yE
zkki
y
yH
xE
zkki
x
zz
z
zz
z
zz
z
zz
z
kH
kH
kE
kE
)(
)(
kHjHiHi
EEE
kji
kEjEiEi
HHH
kji
zyx
zyx
zyx
zyx
zyx
zyx
yE
x
E
z
xE
xzy
yzyE
x
yH
x
H
Z
xH
xzy
yzyH
x
xy
z
z
xy
z
z
Hi
EikHi
EikHi
Ei
HikEi
HikEi
Maxwell’s equations (no sources)
)(
)(
)(
)(
22
22
22
22
yH
zxE
kki
y
xH
zyE
kki
x
xH
yE
zkki
y
yH
xE
zkki
x
zz
z
zz
z
zz
z
zz
z
kH
kH
kE
kE
to solve EZ 、 HZ
TE: Ez= 0, to solve Hz
TM: HZ= 0, to solve Ez
Boundary conditions
00,0:,0
00,0:,0
,0
,0
byyyH
zx
axxxH
zy
z
z
EEby
EEax
Helmholtz equation
022 ZZ HkH
TE
0)()(
0)()(
22
2
22
2
yYky
yY
xXkx
xX
y
x
Separated PDEGeneral solution
)sincos(
)sincos(
)()(),(
22
11
ykDykC
xkDxkC
yYxXyxH
yy
xx
z
Separate variableZik
zzeyYxXzyxH )()(),,(
zikb
na
mmnz
zeyxHH )cos()cos(
Amn = C1C2
,...2,1,0,
,...2,1,0,
00 2010
nb
nK
y
H
ma
mK
x
H
Dy
HD
x
H
ybyz
xaxz
yz
xz
zik
bn
am
mnz
zikb
na
mmnb
nkk
ky
zikb
na
mmna
mkk
kx
zikb
na
mmna
mkky
zikb
na
mmnb
nkkx
z
z
z
z
z
z
z
z
z
z
z
eyxHH
eyxHiH
eyxHiH
eyxHiE
eyxHiE
)cos()cos(
)sin()cos()(
)cos()sin()(
)cos()sin()(
)sin()cos()(
22
22
22
22
TEmn mode
zikb
na
mmnz
zikb
na
mmnb
nkk
ky
zikb
na
mmna
mkk
kx
z
zikb
na
mmna
mkky
zikb
na
mmnb
nKKx
z
z
z
z
z
z
z
z
z
z
z
eyxHH
eyxHiH
eyxHiH
E
eyxHiE
eyxHiE
)cos()cos(
)sin()cos()(
)cos()sin()(
0
)cos()sin()(
)sin()cos()(
22
22
22
22
Fields
222222 )()(b
n
a
mkkkkk yxz
Propagation constant:
22
22
)()(2
1
22 b
n
a
mkkkf
yx
cmn
Cutoff frequency:
TEmn mode
0
)cos(
)cos()sin()(
)sin(
10
10
10
yzx
ikzaz
ikzb
naa
max
ikza
ay
HEE
exHH
eyxHiH
exHiE
xx x
x xx
xx
x
x
xxx xx
xx
xPropagation direction
’
TE10 mode (the fundamental mode)
Phase velocity:dielecpp v
k
bn
am
k
v _222
1
)()(
k
H
E
H
EZ
x
y
y
xTE Wave impedance:
0
)sin()cos()(
)cos()sin()(
)sin()sin(
)cos()sin()(
)sin()cos()(
22
22
22
22
z
zikb
na
mmna
mkky
zikb
na
mmnb
nkkx
zikb
na
mmnz
zikb
na
mmnb
nkk
Ky
zikb
na
mmna
mkk
kx
H
eyxEiH
eyxEiH
eyxEE
eyxEiE
eyxEiE
z
z
z
z
z
z
z
z
z
z
z
fields 222222 )()(b
n
a
mkkkkk yxz
Propagation constant:
22
22
)()(2
1
22 b
n
a
mkkkf
yx
cmn
Cutoff frequency:
TMmn mode
0
)sin()cos(
)cos()sin(
)sin()sin(
)cos()sin(
)sin()cos(
11
11
11
11
11
2
2
2
2
z
ikzb
naak
my
ikzb
nabk
nx
ikzbaz
ikzbabk
ny
ikzbaak
mx
H
eyxEiH
eyxEiH
eyxEE
eyxEiE
eyxEiE
c
c
c
c
TM11 mode (the lowest mode)
Phase velocity:dielecpp v
k
bn
am
k
v _222
1
)()(
kH
E
H
EZ
x
y
y
xTM
Wave impedance:
22 )()(b
n
a
mkc
Mode patterns-- Rectangular waveguide
2.5 Circular waveguide
Mode patterns _ Circular waveguide
2.6 Surface waves on a grounded dielectric slab
Surface waves
• a field that decays exponentially away
from the dielectric surface
• most of the field contained in or near the
dielectric
• more tightly bound to the dielectric at
higher frequencies
• phase velocity: Vdielectric < Vsurface < Vvacuum
Geometries:
2.7 Stripline
Stripline as a sort of “flattened out” coaxial line. Stripline is usually constructed by etching the center conductor to a grounded substrate of thickness of b/2, and then covering with another grounded substrate of the same thickness.
Propagation constant :
with the phase velocity of a TEM mode given by
Characteristic impedance for a transmission:
CvC
LC
C
LZ
p
10
Laplace's equation can be solved by conformal mapping to find the capacitance. The resulting solution, however, involves complicated special functions, so for practical computations simple formulas have been developed by curve fitting to the exact solution. The resulting formula for characteristic impedance is
with
Inverse design When designing stripline circuits, one usually needs to find the strip width, given the characteristic impedance and permittivity. The inverse formulas could be derived as
Attenuation loss (1) The loss due to dielectric filler.
(2) The attenuation due to conductor loss (can be found by the perturbation method or Wheeler's incremental inductance rule).
with
)/(2
tanmNp
kd
t thickness of strip
2.8 Microstrip
For most practical application, the dielectric substrate is electrically very thin and so the field are quasi-TEM.
Phase velocity: Propagation constant:
Effective dielectric constant:
Formula for characteristic impedance: (numerical fitting )
Inverse waveguide design with known Z0 and r:
Considering microstrip as a quasi-TEM line, the attenuation due to dielectric loss can be determined as
Filling factor:
The attenuation due to conductor loss is given approximately
For most microstrip substrates, conductor loss is much more significant than dielectric loss; exceptions may occur with some semiconductor substrates, however.
which accounts for the fact that the fields around the microstrip line are partly in air (lossless) and partly in the dielectric.
Attenuation loss
2.9 Wave velocities and dispersion
• The speed of light in a medium• The phase velocity (vp = /)
)/1(
So far we have encountered two types of velocities:
Dispersion: the phase velocity is a frequency dependent function.
Group velocity: the speed of signal propagation (if the bandwidth of the signal is relatively small, or if the dispersion is not too sever)
The “faster” wave leads in phase relative to the “slow” waves.
• Consider a narrow-band signal f(t) and its Fourier transform:
• A linear transmission system (TL or WG) with transfer function Z(w)
Z()F() Fo()
)()()( FZFo
• If and (ie., a linear function of ),
))()(( jzj eZAeZ
.)( consAZ a
).()(2
1)( )( atAfdeAFtf atj
o
f(t) is a replica of f(t) except for an amplitude factor and time shit. A lossless TEM line (=/v) is disperionless and leads to no signal distortion.
Input Output
Transfer
• Consider a narrow-band signal s(t) representing an amplitude modulated carrier wave of frequency 0:
})(Re{cos)()( 00
tjetfttfts
)()()( 00
Fdteetfs tjtj
• The output signal spectrum:zjeAFs )()( 00
m << 0
Assume that the highest frequency component of f (t) is m, where m << 0 .The Fourier transform is
The output signal in time domain:
deFAdeSts ztjtjo
m
m
)(00 )(Re
2
1)(Re
2
1)(
0
0
For a narrowband F(), can be linearized by using a Taylor series expansion about 0:
...)(2
1)()()( 2
0
2
00 020
d
d
d
d
From the above, the expression for so(t):
)cos()'()( 0000 ztztAfts (a time-shifted replica of the original envelope s(t).)
The velocity of this envelope (the group velocity), vg:
0
1)('
1
d
dvg
2.10 Summary of transmission lines and waveguidsComparison of transmission lines and waveguides
2.10 Summary of transmission lines and waveguidsOther types of lines and guidesRidge
waveguideDielectric waveguide
Slot line
Coplanar waveguide
Covered microstrip
(electric shielding or physical shielding)
(TE or TM mode, mm wave to optical frequency, with active device)
(quasi-TEM mode, rank behind microstrip and stripline)
(Quasi-TEM mode, useful for active circuits)
(lower the cutoff frequency, increase bandwidth ad better impedance characteristics)
Homework1. An attenuator can be made using a section of waveguide operating below cutoff, as shown below. If a =2.286 cm and the operating frequency is 12 GHz, determine the required length of the below cutoff section of waveguide to achieve an attenuation of 100 dB between the input and output guides. Ignore the effect of reflections at the step discontinuities.
2 Design a microstrip transmission line for a 100 characteristic impedance. The substrate thickness is 0.158 cm, with r = 2.20. What is the guide wavelength on this transmission line if the frequency is 4.0 GHz?